Problem 37 - Precalculus with Limits

Question

In Exercises \(33-42,\) find the standard form of the complex number. Then represent the complex number graphically. $$ \frac{9}{4}\left(\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right) $$

Step-by-Step Solution

Answer

The standard form of the complex number is \(-\dfrac{9\sqrt{2}}{8} + i\ \dfrac{9\sqrt{2}}{8}\), and its graphical representation is a point at (-1.27, 1.27) on the complex plane.

Step 1: Identify the magnitude and the angle

The magnitude, denoted by \(r\), is \(\dfrac{9}{4}\), and the angle θ, both in radians, is \(\dfrac{3\pi}{4}\).

Step 2: Apply trigonometric identities

Now, using the identities of the cosine and sine, we calculate the real part \(a = r\cos(θ) = \dfrac{9}{4}\cos\(\dfrac{3\pi}{4}\) = -\dfrac{9\sqrt{2}}{8}\) and the imaginary part \(b = r\sin(θ) = \dfrac{9}{4}\sin\(\dfrac{3\pi}{4}\) = \dfrac{9\sqrt{2}}{8}\).

Step 3: Write in standard form

The standard form of a complex number is \(a + bi\). Substituting the calculated values, we get \(-\dfrac{9\sqrt{2}}{8} + i\ \dfrac{9\sqrt{2}}{8}\).

Step 4: Graphical representation

On the complex plane, plot a point at \((-1.27, 1.27)\) which are the calculated real part (x-coordinate) and the imaginary part (y-coordinate).